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Risk Preferences and Risk Pooling in Networks:Theory and Evidence from Community Detection in Ghana
By Daniel Putman∗
INCOMPLETE AND PRELIMINARY:DO NOT CITE OR CIRCULATE
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Do individuals share risk with others who have similar risk prefer-ences or different risk preferences? That is, do individuals assorta-tively match on risk preferences? I investigate this question in bothpreferred risk sharing networks and effective risk risk pooling com-munities. Using risk sharing network data in Ghana, I construct apreferred network and import community detection (clustering) al-gorithms from network science to partition risk sharing networksinto risk pooling communities. I estimate that individuals pre-fer to assortatively match on risk preferences in risk sharing net-works. While assortative matching holds in risk pooling communi-ties, the magnitude is reduced as compared to earlier estimates. Inother words, risk pooling communities feature more diverse prefer-ences than network neighborhoods. How does this impact welfare ofagents in the network? I build a theoretical model of covariate riskpooling with heterogeneous types from a baseline of perfect idiosyn-cratic risk sharing. Constructing a planners problem, I comparethe actual allocation of types in communities to four benchmarks:optimal risk sharing, no covariate risk sharing, random allocationof types, and preferred allocation of types.JEL: O12, 016, 017, L14Keywords: Risk Sharing, Networks
∗ University of California, Davis Ag. & Resource Economics. This paper has benefited immensely fromconversations with Travis Lybbert, Ashish Shenoy, Laura Meinzen-Dick, Thomas Walker, Tor Tolhurst,Arman Rezaee, as well as other comments from UC Davis Development Tea. All errors herein are myown.
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I. Introduction
Formal insurance markets are often missing in developing economies (Mccord
et al., 2007). In the absence of formal insurance markets, informal risk sharing,
mediated through social networks, helps fill this void (Fafchamps, 2003, 2008).
Informal risk sharing networks have often been studied as a way to smooth id-
iosyncratic risk, or risk uncorrelated with that faced by others (Townsend, 1994;
Kinnan, 2014). This includes risks like illness, loss of employment, and theft: if
you lose your job, I insure you; If I lose my job, you insure me. In contrast,
the use of these networks to smooth covariate risk has received less attention.
This includes price and weather risk, which are substantial. Despite this, when
risk preferences differ, sharing covariate risk can lead to welfare improvements by
shifting covariate risks from more risk averse to less risk averse agents in exchange
for a “premium” (Chiappori et al., 2014). However, the ability to share covariate
risk depends crucially on the ability of agents with different risk preferences to
pool risk.
This suggests an empirical question of interest: do individuals share risk with
others who have similar risk preferences or different risk preferences? That is,
do individuals assortatively match on risk preferences? We can split this into
two sub-questions. First, the process of sharing risk is mediated through social
connections and and relies on bilateral transfers ex post. Looking at bilateral risk
sharing relationships, do individuals prefer to share risk with others with similar
risk preferences? Second, despite the bilateral nature of transfers, the insurance
derived from risk sharing networks extends beyond one’s immediate social con-
nections. When we consider the effective unit of risk pooling, do individuals pool
risk with similar types or different types?
These first two questions speak to a literature on risk sharing in networks,
RISK AVERSION IN NETWORKS 3
exemplified by Attanasio et al. (2012). Attanasio et al. (2012) experimentally
demonstrate that individuals prefer to share risk with those who have similar risk
preferences. I confirm this earlier experimental result in a observational risk shar-
ing network. Using a subgraph generation model (or SUGM) estimated using risk
sharing network data in Ghana, I estimate that individuals prefer to assortatively
match on risk preferences. That is, they prefer to match with individuals who
have a similar degree of risk aversion. This effect is driven by less risk averse
individuals, who tend to connect to their own type. However, more risk averse
agents are also more likely to connect to less risk averse agents than more risk
averse agents.
The second question goes beyond bilateral risk sharing. To quantify the effective
breadth of risk pooling, I take advantage of algorithms to find tightly knit groups
within these risk sharing networks. Specifically, I import community detection
(read: clustering) algorithms from network science to partition risk sharing net-
works into risk pooling communities (Newman, 2011). I treat these communities
as the relevant unit of risk sharing at the subvillage level - large enough to encom-
pass all relevant agents for risk pooling, small enough not to include individuals
who are distant in the network. Re-estimating the SUGM on a network aris-
ing from these communities, I find that while assortative matching holds in risk
pooling communities, the magnitude is reduced as compared to earlier estimates.
In other words, risk pooling communities feature more diverse preferences than
network neighborhoods.
Given these findings, what are the welfare impacts of the degree of (and reduc-
tion in) assortative matching? Similar to Chiappori et al. (2014), I investigate
the welfare implications of covariate risk pooling with heterogeneous risk pref-
erences. However, in contrast, I look at risk pooling at the community level as
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opposed to the village level. With the introduction of subvillage risk sharing
communities, it matters a great deal who pools risk with whom. To demonstrate
the importance of how types are distributed, I divide individuals into more and
less risk averse types and quantify the difference in the actual allocation of types
in communities relative to some benchmarks: (1) the case where communities
are entirely assortatively matched (a worst case scenario from the perspective of
covariate risk pooling), (2) the case where assortative matching does not differ
between bilateral (preferred connections) and communities, (3) randomly sorted
communities, and (4) optimal communities as characterized by a model of risk
pooling. Furthermore, I quantify the welfare gains or losses relative to these four
benchmarks.
Finally, how do we explain the differences between bilateral (or preferred) risk
sharing and the community level (or effective) risk pooling? To square preferences
with effective risk pooling groups, I use Subgraph Generation Models, which
treat network structure as an emergent property of subgraph formation. Several
mechanisms can account for the difference. For example, using a simple rule
of “close matches,” individuals might form chains of agents with gradients of
risk preferences. These allow for agents of distant risk aversion to be present
in the same community. Another mechansim is that while most agents prefer
to link to agents of similar risk aversion, central agents (perhaps shopkeepers
or moneylenders) may be willing to link to all types, meaning agents of various
types are relatively close to many individuals within the network. In general,
a mechanism where communities are more diverse than bilateral relationships
depends on some emergent property. To my knowledge, this paper is the first to
investigate this emergent property feature of risk sharing networks.
Following this introduction, section II [provides background, including litera-
RISK AVERSION IN NETWORKS 5
ture]. Section III builds a theoretical model of risk pooling in communities with
heterogeneous risk preferences. Section IV describes the dataset I use and the
construction of the networks. Section V details the empirical strategies used and
section VI presents these results. Finally, section XX concludes.
II. Background
A. Graphs and Risk Sharing Networks
Throughout this paper, I will draw on graph theory to describe intuition, for-
malize intuition and visually represent risk sharing networks. A graph g is a set of
nodes and an edgelist (which naturally contains edges). I refer to these nodes and
edges by their subscripts. I subscipt nodes by i. For edges, I use the combination
of subscripts i and j to refer to that edge: if there is a connection between i and
j, I say ij ∈ g, hence ij is in the edgelist. An adjacency matrix represents these
nodes and edges in an n × n matrix A = A(g). For the scope of this paper, I
work with unweighted and undirected graphs, choosing to work with reciprocal
relationships. Thus aij = 1 if ij ∈ g and 0 if not. The adjacency matrix is also
symmetric: aij = aji for all i, j. The diagonal aii = 0 by construction.
Nodes and edges go by many other names. In the case of risk sharing, nodes
represent agents and edges represent the social connections between those agents.
I will use “agents” and “individuals” interchangeably when referring to nodes in
the network. Likewise, I will use “links” and “connections” interchangeably when
referring to edges. Dyads are not interchangeable, however: dyads are all possible
combinations ij regardless of whether that edge exists in the network.
How do risk sharing networks relate to risk sharing arrangements? I take the
view that risk sharing arrangements (and by extension, risk pooling arrange-
ments) are ex ante informal contracts specifying transfers based on ex post states
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of the world. Given the complexity of such a contract, the edge itself does not tell
the story of the risk sharing arrangement happening over that edge. For param-
eterized contracts, however, we can list these parameters as edge characteristics.
Depending on whether the risk sharing network is ex ante or ex post, these social
connections may reflect agreements of transfers or past transfers (or some mix of
the two).
Some node level measures I will use are neighborhood and degree. An agent i’s
neighborhood is all agents j such that ij ∈ g. An agent i’s degree is the size of
agent i’s neighborhood. We can compute degree d(i) =∑n
j=1 aij . On the network
level, some measures of network structure are density and clustering. The density
of a network is the total number of connections divided by the number of potential
connections. We can compute density
(1) density(g) =1
2(n2
) n∑i=1
n∑j=1
aij
The clustering coefficient of a network takes (three times) the number of triangles
and divides it by the potential number of triangles. We can compute clustering
(2) clustering(g) =1
2(n3
) n∑i=1
n∑j=1
n∑k=1
aijajkaik
Other network concepts that arise will be explained as they arise.
B. Related Literature
Attanasio et al. (2012) uses a risk pooling experiment to test for assortative
matching in 70 municipalities in Colombia. In this study, the authors measure
risk aversion by offering a set of gambles. Then they give their experimental
RISK AVERSION IN NETWORKS 7
subjects the ability to form risk pooling groups to pool risk. They find close
friends and relatives match assortatively on risk attitudes, whereas less familiar
dyads tend not to match at all. My paper provides a scientific replication of
the results of Attanasio et al. (2012) using observational data from a different
context.1
Chiappori et al. (2014) Structurally estimates risk aversion coefficients in vil-
lages when risk sharing arrangements are complete. My paper differs in modeling
approach by using CARA preferences as opposed to constant relative risk aver-
sion. However, the solution of the two modeling approaches share an intuition.
Individuals who have risk preferences near to risk neutral make a market in taking
on additional covariate risk.
III. Theoretical Model
Consider a case where a risk neutral planner constructs a community in a village
to maximize expected utility within a community. Here I leave aside community
size and its impact on community composition and focus on optimal community
composition. In this case, we think about community composition with regard to
risk aversion, with relatively less and more risk averse individuals. We set up this
problem in two steps. First, we characterize how risk is shared in a community
according to its composition. Second, using the solutions and value functions from
the first optimization problem, we write a planners problem maximizing aggregate
expected utility of consumption in a village with communities conditional on the
composition of those communities.
1Looking at the replication data, it is interesting to note that the size of these risk pooling groupsdiffer in size considerably. In addition, by “visual inspection” of the replication data provided withthe paper, chosen groups are definitely (strongly, though incompletely) related to detected communities(both Walktrap and Edge betweenness) - how can this be quantified?
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A. Risk Sharing in Communities
To model risk sharing in communities, we start from a baseline of perfect id-
iosyncratic risk sharing. This means that all shocks that are above and below a
villager’s mean income are smoothed to their mean income (we will assume these
are zero for the purposes of this problem). After this set of transfers takes place,
a round of risk shifting takes place. Less risk averse individuals may take on more
of covariate risk. This covariate risk derives from both the average idiosyncratic
shock, which in general is not zero, and the (perfectly correlated) covariate shock.
More risk averse agents are able to take on less of the covariate risk over time,
shifting them on to less risk averse individuals. However, less risk averse indi-
viduals are still risk averse, so they require some compensation for the risk they
take on. Thus, recurring transfers are made to these individuals regardless of the
covariate shock.
Setup. — Suppose a community of fixed size N . Villagers have exponential
utility functions with coefficient of absolute risk aversion η.
u(c) = −e−ηc
However, suppose we have low and high risk aversion households, where type
and indexed by ` = 1, 2. Then we assume η2 > η1 > 0. N` is number of
individuals of type `, and p = N1/N characterizes the composition of the group.
All households face a perfectly correlated shock, yv and an idiosyncratic shock
yi. Risk is symmetric between households and between types: Household level
RISK AVERSION IN NETWORKS 9
shocks, yi ∼iid N(0, σ2) and type level shocks yv ∼iid N(0, ν2).
y`i = yi + yv
Taking account of risk sharing process, we write the consumption of household
i of type ` as a weighted sum of the idiosyncratic and covariate shock in the
community. For type ` = 1
c1i =
(θ
p
)(1
N
N∑i=1
yi + yv
)− λ1i
and for type ` = 2
c2i =
(1− θ1− p
)(1
N
N∑i=1
yi + yv
)− λ2i.
The proportion of covariate risk that is borne by the less risk averse individuals
in the community is regulated by the parameter θ ∈ [0, 1]. When θ = 1, all risk is
taken on by less risk averse individuals, when θ = p, risk is shared equally among
all members of the community (i.e., only idiosyncratic risk is pooled), and when
θ = 0 all risk is taken on by more risk averse households. Conversely, λ`i regulates
the recurring transfers from the more risk averse to the less risk averse. Thus, it
is the case that the aggregate transfer into the pot exceeds the aggregate transfer
out:
−N1λ1i ≤ N2λ2i.
Due to the exponential utility function and normal distribution of shocks, we
are able to represent expected utility as a mean-variance decomposition
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E(U(c`i)) = E(c`i)−η`i2V ar(c`i)
We will refer to the special case of EU0 = E(U(c1i|θ = p))), which simply de-
scribes the utility from idiosyncratic risk sharing in absence of risk sharing.
Optimization Problem. — We maximize expected utility of less risk averse
agents subject to several constraints. Constraint 4 is an incentive compatibility
constraint: more risk averse agents cannot being worse off than in the case where
they only perfectly pool idiosyncratic risk. Constraints 5 and 6 serve as individual
budget constraints for each type, and finally 7 serves to ensure feasibility of the
recurring transfers.
maxλ1,λ2,θ
E(U1(c1i))(3)
subject to EU0 ≤ E(U(c2i))(4)
c1i =
(θ
p
)(1
N
N∑i=1
yi + yv
)− λ1i(5)
c2i =
(1− θ1− p
)(1
N
N∑i=1
yi + yv
)− λ2i(6)
0 ≤ pλ1 + (1− p)λ2(7)
Solutions and Value Functions. — How much covariate risk is shifted to the
less risk averse agents? Recall, if θ = 1, all covariate risk shifts to less risk
averse inviduals, and if θ = p, the baseline of perfect idiosyncratic risk sharing is
RISK AVERSION IN NETWORKS 11
maintained.
(8) θ∗(p) =pη2
(1− p)η1 + pη2
Since η2 > η1, θ∗ > p. This means some degree of covariate risk is shifted to less
risk averse individuals. Likewise, unless η1 = 0 (we assumed it doesn’t) or p = 1
not all risk is taken on by the less risk averse. This brings us to an important
result of the model: group composition matters for the degree of risk sharing.
What are more risk averse agents willing to pay to shift risk away? Since λ∗2 is
paid into the community pot, Type 2’s willingness to pay depends on their own
risk aversion, type 2’s risk aversion, and group composition
(9) λ∗2(p) = −η2
2
(1− η2
1
((1− p)η1 + pη2)2
)
where the expression in parentheses lies between 0 and 1. Because risk is sym-
metric in this model, the transfer does not depend on covariate risk. Presumably,
were this not the case, it would appear in the above solutions.
Finally, type 1 will maximize their utility and hence the payments they receive
from type 2. We can write λ∗1 by converting type 2’s willingness to pay into type
1’s average payment:
(10) λ∗1(p) = −1− pp
λ∗2(p).
Once we have these solutions we are able to compute the value functions we
will use in order to solve the planner’s problem. These are
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V1(p, η1, η2) =η1
2
(1− pp
)(1−
(η1
(1− p)η1 + pη2
)2)
(11)
− η1
2
(η2
(1− p)η1 + pη2
)2(σ2
n+ ν2
)
(12) V2(p, η1, η2) =η2
2
(1 +
(η1
(1− p)η1 + pη2
)2(1 +
(σ2
n+ ν2
)))
B. Planner’s Problem
Setup. — The risk neutral planner seeks to maximize aggregate expected utility
of consumption conditional on the composition of communities. We have two
communities, g = A,B. We will update the notation from the first stage slightly.
For a given community g, Ng is the community size and NA + NB = N . Then
N`g is the number of individuals of type ` in community g and p`g =N`gNg
.
Optimization Problem. — We can state the planner’s problem as an optimiza-
tion problem where the planner
maxN1A
N1AV1(p1A) + N2AV2(p1A) + N1BV1(p1B) +N2BV2(p1B)(13)
subject to N1 = N1A +N1B(14)
N2 = N2A +N2B(15)
NA = N1A +N2A(16)
NB = N1B +N2B(17)
RISK AVERSION IN NETWORKS 13
To simplify this problem, we consider the simple case where we have an equal
number of more and less risk averse types. That is, N1 = N2. This implies that we
can encompass the entire problem just by looking at one choice parameter, p1A,
and conditioning on the size of the smaller community, NA. p1A = N1ANA
, and we
can express p2A = 1− p1A, p1B = 2N1BN = 2(N1−N1A)
N and p2B = 1− p1B. Setting
N1 = N2 reduces the set of constraints to three, and our quick computations takes
account of these three constraints. Hence we have the problem
maxN1A
N1AV1
(N1A
NA
)+N2AV2
(N1A
NA
)(18)
+N1BV1
(2(N1 −N1A)
N
)+N2BV2
(2(N1 −N1A)
N
)
Solving this planners problem for an analytic solution is relatively difficult.
However, it is relatively easy to characterize the optimal allocation of types numer-
ically. In Figure 1, I plot the objective in Problem 18 against p1A, the new choice
variable. To construct this example, I set σ2c = 1002, N = 100, N1 = N2 = 50,
and set η1 ≈ 0.0016, η2 ≈ 0.0037, the means of these for the type 1 and type 2
in my data (see Section IV.A for construction of coefficients of risk aversion and
types).
Looking at Figure 1, we see that welfare is maximized when p1A = 0.5, when
diversity of types is maximized. Likewise, we see that welfare is minimized when
as p1A approaches 0 or 1, when diversity of types is minimized. Interestingly, un-
equal sized suboptimal communities improve over more equally sized suboptimal
communities holding p1A equal. Also interesting, welfare is not symmetrically
suboptimal when the proportion of less risk averse agents strays from zero. If
you overfill communities with a type (i.e., p1A 6= 0.5) it is better to “overfill” the
smaller community with type 1 (less risk averse) agents as opposed to overfilling
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−1250
−1200
−1150
−1100
0.00 0.25 0.50 0.75 1.00Proportion Less Risk Averse (Type 1) in Smaller Community
Tota
l Los
s du
e to
Ris
k (L
CU
)
Size of smaller community
3035404550
Figure 1. Optimal Allocation of Types Between Unequally Sized Communities
the larger community. For another way to look at this, in the appendix I plot the
proportion of risk taken on by both groups in a risk pooling frontier (see Figure
D1)
IV. Data and Context
A. Data
The data I use comes from AMA-CRSP Survey of Risk Coping and Social
Networks in Rural Ghana. In particular, these data feature information about
assets, income, consumption (positive and negative) shocks data for households in
four villages in Ghana. The survey ties this to network data collected to measure
risk sharing networks (Walker, 2011a). The survey instrument and technical
details can be found in Walker (2011b).2
2Walker (2011a) and Walker (2011b) as well as the raw data can be downloaded athttp://barrett.dyson.cornell.edu/research/datasets/ghana.html
RISK AVERSION IN NETWORKS 15
Risk Loving (Maximum)Risk NeutralRisk Averse (Moderate)Risk Averse (Maximum)Not Surveyed
Risk Loving (Maximum)Risk NeutralRisk Averse (Moderate)Risk Averse (Maximum)Not Surveyed
Figure 2. Risk Sharing Networks with Risk Aversion Plotted: Darmang (top) and Pokrom
(bottom)
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Risk Loving (Maximum)Risk NeutralRisk Averse (Moderate)Risk Averse (Maximum)Not Surveyed
Risk Loving (Maximum)Risk NeutralRisk Averse (Moderate)Risk Averse (Maximum)Not Surveyed
Figure 3. Risk Sharing Networks with Risk Aversion Plotted: Oboadaka (top) and Konkonuru
(bottom)
RISK AVERSION IN NETWORKS 17
Risk Preferences. — Leaving aside risk neutrality, using a set of four hypo-
thetical gambles we use exponential preferences to measure individuals risk aver-
sion.The first two menus presented are in the gains domain. In the first menu,
the risky gamble YB is held fixed while increasing the sure payment. In the sec-
ond, the sure payment is held fixed while reducing the upside of the gamble. We
assume that if an individual reaches a point of indifference between two gambles,
we assign them to the midpoint between the two gambles. Hence, if the mean
differs, we take the average of the mean of the two gambles and assign this value
to the point of indifference. If the variance differs, we take the average of vari-
ances and assign this value to the point of indifference. The second two menus
are reflections of the first two onto the domain of losses.
(19) u(c) =1− e−ηc
η
For mathematical tractability, we assume YB is normally distributed. Using this,
we represent expected utility as a mean variance utility function (Sargent, 1987).
We write
(20) E(Y )− η
2V (Y )
I look at a menu of choices between hypothetical gambles. YA is a set of sure
payments and YB is a set of risky income sources. Respondents are able to choose
between these will be indifferent between the two when
(21) E(YB)− η
2V (YB) = YA
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Observing this indifference point, we can write
(22) η =2(E(YB)− YA)
V (YB)
and recover the coefficient of absolute risk aversion. We compute η for each lottery
and individual. To combine these into measures of risk aversion, we average over
individuals and use the mean risk aversion.
Risk Sharing Network. — To construct the risk sharing network, I use the
intersection of a gift network and a trust network. To construct the gift network,
a link occurs if invididual i has received a gift from j and also if i has given a
gift to j. In the trust network, I report a link if both i and j report trusting each
other (in previous uses of this data, this has been a “strong ties” network). If a
tie occurs in both networks, I record a tie between i and j and use this as my risk
sharing network.
For an alternative network, I aggregate the trusted ex post gift network and a
trusted kin network. In the kin network a link occurs if i reports being related to
j or vice-versa and the trust network is constructed as above. To aggregate these
networks, I take the intersection of the trust and the kin networks first. Then,
I include a link in the risk sharing network if a link occurs between i and j in
either the trusted kin or the trusted gift network.
V. Empirical Strategy
A. Community Detection
Intuition. — Consider a potential risk sharing process. A large gift is given to a
randomly chosen individual in a risk sharing network. The household gives a gift
RISK AVERSION IN NETWORKS 19
to a (network) neighboring individual who is less well off than they are, sharing
their positive shock equally. Ex ante, all the individual’s neighbors should be
equally likely to receive this gift. Having received this gift, this individual is
also is obligated to share with their worse off neighboring individual, provided
they are not worse off than all of their network neighbors. This process of risk
sharing continues. Eventually, the individual receiving the transfer ends with
an effective shock which is still worse than their neighbors and gives no gifts.
We end up with progressively smaller transfers “walking” randomly through the
network. Individuals who are close to the individual who received the prize will
have a high probability of receiving a transfer, while those far will have a low
probability. Likewise, even if one does not receive a transfer in the first step, if
an individual is connected to many of the same households as the one receiving
the prize, they get additional chances for a gift. We would expect that within a
dense and clustered portion of the network, most often the gains from the positive
shock will not make it far, instead getting “trapped” in the local network.
Similarly, we can present this same intuition for negative shocks: a randomly
chosen individual a negative shock. Based on a similar rule, they might request
a favor from one of their neighbors, who might request a favor from one of their
neighbors, and so on. We end up with progressively smaller transfers being drawn
through the network. Close individuals will more likely receive a request for a
gift. Likewise, requests will remain trapped within dense and clustered portions
of the network.
Walktrap Algorithm. — This process functions as a way risk is effectively
pooled and mirrors the intuition of the walktrap algorithm. The walktrap algo-
rithm uses random walks over the network to measure the similarity of nodes.
The algorithm exploits the fact that a random walker in a graph tends to get
20 OCTOBER 2019
“trapped” within tightly knit sections of the graph. A random walker starts at
a random node i and moves to an adjacent node with probability 1/d(i) (where
d(i) is the degree of i). Then for each of a fixed number of periods, this process is
repeated from the landing node, k, moving to an adjacent node with probability
1/d(k). If nodes are in the same community a random walk of fixed length from
node i should often land on node j.
However, walks from node i frequently landing on node j is not a perfect con-
dition for nodes to reside within the same community. First, nodes with high
degree tend to receive more random walks, so far off individuals with high degree
might be labeled as within the same community. For this reason, the measure
of distance should take into account the degree of the receiving node. Second,
individuals in the same community should see other nodes in the the network
similarly. Pons and Latapy (2004) uses this second fact to build a measure of
similarity.
(23) rij =
√∑nk=1(P sik − P sjk)2
d(k)
where P sik is the probability of a random walk of length s from node i landing
on node k. The algorithm works by iteratively merging nodes who are similar
into communities.
The algorithm starts by assigning each node into its own community in the
network. Similarities are computed between adjacent nodes using the measure
above and the closest two communities are merged. Then, the merged community
remains in the network as a single node (with all its out of community links
preserved). Similarities are updated and the process repeats until all nodes are
RISK AVERSION IN NETWORKS 21
merged into one community. This builds a dendrogram, or a tree depicting the
merges of nodes into communities. Each merge is depicted by a split in the
tree. This gives us many possible community assignments. To choose an optimal
community assignment, a tuning statistic called modularity is used to cut the
dendrogram at one of the splits.
Modularity measures the internal quality of the community by looking at how
many links exist within the community compared to a how many we would expect
at random. The measure follows from a thought experiment: suppose you were
to take a graph and randomly “rewire” it. This rewiring preserves the degree of
individual nodes, while destroying the community structure. We take the average
number of within community links from rewiring as the counterfactual. Having
many more links within the community than the counterfactual implies a good
community detection, fewer implies a poor community structure. For more about
modularity, including the formal definition, please see Appendix B.B2.
Networks Arising From Community Detection. — After we have assigned
nodes to communities, we construct an additional risk sharing network using these
community assignments. Assuming that effective risk pooling takes place at the
community level, all nodes assigned to a given community are linked within the
network. We represent the community graph using an adjacency matrix C where
cij = 1 if i and j are in the same community and 0 if not. Like the adjacency
matrix, C is symmetric. I use this network in dyadic regression and subgraph
generation models to measure who you pool risk with in effective terms.
B. Dyadic Regression
Before diving into the Subgraph Generation Models, it is useful to estimate
dyadic regressions. While dyadic regressions cannot handle “higher order” fea-
22 OCTOBER 2019
tures like stars or triangles, they are familiar, interpretable as regressions, and
comparable with past literature. In particular, I note similarities and differences
between these results and those found in Attanasio et al. (2012).
In these regressions, each pair of nodes is treated as an observation. Let aij = 1
if individual i has given and received gifts from j and both i and j trust each
other. Hence our dependent variable is the risk sharing network defined by using
trusted gift givers. This construction requires that aij = aji so the adjacency
matrix A is symmetric. In addition, as required by Fafchamps and Gubert (2007)
all explanatory variables also enter symmetrically. In particular, I estimate a set
of four linear probability models. The most parsimonious model regresses risk
sharing connections on differences in measured risk aversion,
(24) aij = β0 + β1|ηi − ηj |+ εij
where ηi is the risk aversion of individual i and εij is the error term. Here (as
in the following specifications) a negative estimate β1 is evidence of assortative
matching, i.e., that individuals prefer to share risk with individuals who have
similar risk preferences to their own. In our sample, less risk averse agents tend
to be more well connected and have higher degree relative to more risk averse and
risk loving agents. A second specification includes the sum of degrees di and dj
to control for the correlation between risk aversion and degree.
(25) aij = β0 + β1|ηi − ηj |+ β2(di + dj) + εij
A positive estimate of β2 suggests that individuals who have more total links are
more likely to link to each other. Given this endogenous construction, positive
RISK AVERSION IN NETWORKS 23
estimate will not come as a surprise and is closer to an exercise in accounting. A
third specification examines kin ties as a predictor of risk sharing connections.
(26) aij = β0 + β1|ηi − ηj |+ β3fij + εij
where family is fij = 1 when i and j report being family members. A positive
estimate of β3 suggest that family are more likely to be connected within the risk
sharing network. A fourth specification combines specifications 1 and 2.
(27) aij = β0 + β1|ηi − ηj |+ β2(di + dj) + β3fij + εij
Finally, a 5th specification introduces interactions between the difference in coef-
ficients of risk aversion and family ties.
aij = β0+β1|ηi − ηj |+ β2(di + dj) + β3fij(28)
+β4fij × |ηi − ηj |+ εij
A negative estimate of β4 is evidence that assortative matching is stronger among
family members. Moreover, if β1 + β4 is negative, this provides evidence that
within family members, risk aversion is an important determinant of risk sharing
connections.
In addition, I re-estimate regressions 24, 25, 26, 27, and 28 with detected com-
munities as the network (as opposed to the network adjacency matrix). In all of
the above specifications, I replace aij with cij , the ijth entry of the community
matrix C. cij = 1 if i and j are in the same detected community, and 0 otherwise.
Degree sum is included in the re-estimation of 25, 27 and 28. While these coef-
24 OCTOBER 2019
ficient estimates are comparable, note that in the community network this ends
up being 2× (community size−1).
As mentioned above, I estimate these regressions as linear probability models
(though results are similar estimating logistic regression). Note however, that
the errors are non-independent. In particular, the residuals of dyads involving i
might be arbitrarily correlated, i.e., Cov(εij , εik) 6= 0. To correct for this non-
independence, I cluster by dyad as proposed by Fafchamps and Gubert (2007).
As a robustness check, we run the same regressions in logistic form (see appendix
for tables).
C. Subgraph Generation Models
Intuition. — A useful tool for understanding risk sharing networks and com-
muntities is called a Subgraph Generation Model (SUGM). SUGMs treat networks
as emergent properties of their constituent subgraphs.3 A subgraph (or induced
subgraph) of a graph is the graph obtained from taking a subset of nodes in the
graph and all edges connecting those nodes to each other. For example, if we
select a subset of two nodes of a graph, the subgraph will be either a link or two
unconnected nodes. If we take three points, we might observe a triangle (a trio
of nodes all connected by edges), a line (one node connected to the two others),
a pair and an isolate (only two nodes connected), or an empty subgraph (three
unconnected nodes). In general, we focus on connected subgraphs for the SUGM.
In three node example above, the means we leave aside the pair and isolate and
the empty subgraph, focusing on the triangle and the line. Likewise, while a link
is an interesting subgraph, two unconnected nodes is not. When describing these
3While Exponential Random Graph Models (ERGMs) have similar motivation, they do not succeedat reconstructing graphs with any success. They depend on an assumption of independence of links, ifthis does not hold they are not consistent. To the contrary, much of the theory of risk sharing wouldexpect links are dependent on each other (Chandrasekhar and Jackson, 2018).
RISK AVERSION IN NETWORKS 25
models, I will use “features” of the SUGM interchangably with subgraph.
While SUGMs can be estimated using GMM, I am able to directly estimate the
parameters using an algorithm given by Chandrasekhar and Lewis (2016) and
Chandrasekhar and Jackson (2018). Estimating a SUGM directly is essentially
estimating the relative frequency of various subgraphs in a network. However,
we can’t stop at simply estimating the features. Because networks are the union
of many subgraphs, subgraphs might overlap and incidentally generate new sub-
graphs. For example, three links placed bewteen ij, jk, and ik would incidentally
generate a triangle. We want to estimate the true rate of subgraph generation.
To do this, I order subgraphs by number of links involved in their construction.
Then, I compute the number of subgraphs generated of that type, but only if
they are not a portion of a “larger” subgraph. For subgraphs of same size, order
is arbitrary, but must exclude occurrences of this subgraph incidentally gener-
ated by other subgraphs who are further along in the order. For example, for
a SUGM featuring links and triangles, I order links first, triangles second, etc.
While counting links and potential links, I neglect pairs of nodes ij if jk and ik
are in the graph.4 This algorithm yields consistent estimates of the true rate that
these subgraphs are generated at.
Links and Isolates Subgraph Generation Model with Types. — I start by
estimating two simple SUGMs on the risk sharing network, using only links and
isolates as features.
The other two SUGMs seek to understand how individuals of different risk
preferences connect to each other. For various reasons, a small subset of indi-
viduals in the network did not participate in the survey module that recovered
4 If we to add lines of 3 nodes, I could order these before or after triangles. Ordering lines beforetriangles I would look at potential links ij and jk where ik is not in the graph. Likewise, I would needto remove pairs of nodes ij if jk or ik are in the graph.
26 OCTOBER 2019
risk preferences.5 Hence, we start with a 2-type model to diagnose the differences
between those individuals who participated and those who did not. I estimate
five features: Isolates of surveyed nodes, isolates of nodes who were not surveyed,
and links within surveyed nodes, links between surveyed nodes and non-surveyed
nodes.
Of those in the network who have a risk aversion coefficient, I split these indi-
viduals into three groups: risk loving, less risk averse, and more risk averse. Risk
loving are those with η < 0. This accounts for about 20% of the networks who
have preferences. I split the remaining risk averse individuals into evenly sized
groups of 40% each, with more risk averse individuals being above a cut-point.
The second model is a full model with the types described above plus a type that
has missing data for a total of four types.
For each model I estimate βn =(βnI,`∀`, βnL,`,`∀`, βnL,`,r∀`,∀r
). Coefficients
for isolates of type l, βnI,` are estimated
(29) βnI,` =
∑ni=1 1(deg(i) = 0|li = `)
n`,
coefficients for within links of type `, βnL,`,` are estimated
(30) βnL,`,` =
∑n−1i=1
∑nj=i+1 aij × 1(li = `)× 1(lj = `)∑n−1
i=1
∑nj=i+1 1(li = `)× 1(lj = `)
,
and coefficients for links between type ` and type r, βnL,l,r are estimated
(31) βnL,l,r =
∑n−1i=1
∑nj=i+1 aij × (1(`i = l)× 1(`j = r) + 1(`i = t)× 1(`j = l))∑n−1
i=1
∑nj=i+1 1(`i = l)× 1(`j = r) + 1(`i = t)× 1(`j = l)
.
5Some of these individuals were not surveyed at all, but appear in the network, others may be partof the sample who missed that particular round or module.
RISK AVERSION IN NETWORKS 27
From proposition C.2 in Chandrasekhar and Jackson (2018) under some sparsity
conditions6, Σ−1/2(βn − βn0 ) → N(0, I) where βn0 is the true rate of subgraph
generation. For a feature `, the variance of the feature is the entry on the diagonal:
(32) Σs,s =βn0,s(1− βn0,s)
κs(nms
)where ms is the number of nodes involved in the feature and κs is number of
different possible relabelings of the feature (note: for both isolates and links
κs = 1). So I estimate the standard errors of the SUGM,
(33) σs,s =
√βns (1− βns )
κs(nms
). For the results, κs
(nms
)is the “sample size” of the feature.
Pooled Subgraph Generation Models. — Since network data I am using has
four networks, I need to make choices as to how to handle these multiple net-
works in the Subgraph Generation Model. One approach would be to estimate a
subgraph generation model for each village and average the coefficients of these.
A different strategy, and one that relies on the same asymptotics as the single
network case from Chandrasekhar and Jackson (2018) is to pool the counts and
potential counts from the villages to estimate a single coefficient across the vil-
lages. I decide on the latter for the ability to lean on the same asymptotic theory
as the single network case. However, a few considerations need to be made. Prin-
6A first note here is that these networks are sparse by the definition of Chandrasekhar and Jackson(2018). If we assume a constant growth rate of the density of links, we have that they are growing at
about n1/3 or less (which is acceptable). For this particular model, none of the features chosen canincidentally generate any other feature. For example, links cannot generate isolates, nor can isolatesgenerate links. Because the second is true, for this particular model the conditions from Chandresekharmay be overkill, so to speak.
28 OCTOBER 2019
cipally, we can’t simply combine the networks and run the SUGM. For example, it
is unlikely that the dyads that would occur between villages would be reasonable
potential dyads. Hence we need to collect counts of features and potential counts
of features in all four villages before combining. Let countsv be the count of some
subgraph s in village v and potentialsv be the potential number of times that
feature could occur. These reflect the numerator and denominator, respectively,
of equations XX, XX, and XX above. We estimate the coefficient associated with
some subgraph s
(34) βs =
∑4v=1 countsv∑4
v=1 potentialsv
This uses only the relevant potentail occurences of the feature. Similarly, when
estimating the standard errors of a feature, we cannot use the same effective
sample size as we would use if we combined the networks. Let nv be the number
of nodes in the village network. If we take κs(∑
nvms
)we would include many
combinations of nodes that in reality could not form the subgraph in question.
Hence we estimate the standard errors the of pooled SUGM
(35) σs,s =
√√√√ βs(1− βs)κs ×
∑4v=1
(nvms
) .
Differences in Assortative Matching. — These SUGM estimates give me a
way to test for assortative matching between preferred and effective risk sharing
networks. To do this, I compare the estimates of the community pooled SUGM to
the estimates risk sharing network SUGMs. Using the estimates from the second
RISK AVERSION IN NETWORKS 29
set of SUGMs, I construct the ratios,
R1,1 =βCL,1
βL,1(36)
R2,2 =βCL,2
βL,2(37)
R1,2 =βCL,1,2
βL,1,2(38)
which correspond to within links for agents of type 1, within links for agents of
type 2 and links between types. Looking at the between links, for example, in
the expression for R1,2, βCL,1,2 is the probability that a pair of agents where i is
of type 1 and j is of type 2 are linked in the community graph. Likewise βL,1,2 is
the probability that this pair of agents link in the risk sharing network. However,
we face the issue that the network arising from community detection tends to be
more dense than the risk sharing network. To construct a baseline to compare
these ratios to, we construct a ratio for all nodes with preferences, from the first
SUGMs.
(39) Rhp =βCL,hp
βL,hp
From this we compute the ratio of ratios. To see how assortative matching differs
between the risk sharing network and the community network, we perform the
following hypothesis tests: H0 : R1,1/Rhp = 1, H0 : R1,1/Rhp = 1, and H0 :
R1,1/Rhp = 1.
Using approximations for the mean and variance of a ratio7, and using param-
eter estimates as stand-ins for their means, we can use the following analytic
7https://www.stat.cmu.edu/ hseltman/files/ratio.pdf
30 OCTOBER 2019
expression for the variance
(40) V ar
(βCL,s
βL,s
)=
(βCL,s
βL,s
)2((σC)2
(βCL,s)2−
2Cov(βCL,s, βL,s)
βCL,sβL,s+
σ2
β2L,s
)
Given that the two coefficients derive from a similar data generating process and
measure a similar quantity, it is intuitive that Cov(βCL,s, βL,s) > 0.8 Maintain-
ing this assumption, it is conservative to estimate the variance of the ratio by
assuming Cov(βCL,s, βL,s) = 0. This assumption leaves us with
(41) V ar
(βCL,s
βL,s
)=
(βCL,s
βL,s
)2((σC)2
(βCL,s)2
+σ2
β2L,s
)
Using the ratios in place of the coefficients, we can also find the variance of the
ratio of ratios, making similar (and similarly conservative) assumptions about
the covariance of ratios. For the test of the ratio of ratios, we use these analytic
expressions to construct the standard errors for the tests above.
(42) V ar
(Rs,rRhp
)=
(Rs,rRhp
)2(V ar(Rhp)
R2hp
+V ar(Rs,r)
R2s,r
)
The reported standard errors are built from these expressions.
D. Welfare Implications of Type Allocation
Scenarios
1) Worst case scenario: communities are entirely assortatively matched
2) No emergent properties: Same degree of assortative matching in network as
8My priors are that the correlations between these two coefficients would be close to one.
RISK AVERSION IN NETWORKS 31
in community
3) Randomly sorted
4) Communities optimally sorted per the model above
For each of the above
• How does the allocation of types differ?
• How does this impact welfare?
VI. Results
A. Community Detection
We assign individuals to risk pooling communities using walktrap community
detection with walks of 4 steps applied to our risk sharing network. I general,
longer walks tend to result larger communities, whereas smaller walks result in
smaller communities (For a demonstration of how communities vary by length of
walk, see Figure D3). For the resulting community detection see Figures 4 and 5.
For each village:
1) Number of communities
2) Size of communities
3) Number of isolates
4) Modularity
[TABLE HERE]
32 OCTOBER 2019
8
8
25
88
8
8
8
8
11
11
11
26
16
16
8
83
17
15
1
8
818
3
3
3
88
1
1
1414
20
20
1919
3
15
2
12
2
2
18
18
20
20
15
8
7
7
3
8
3
8
8
9
9
8
8
3
3
8
8
4
2
8
8
8
6
6
18
88 8
15
15
3
3
99
8
8
12
8
8
13
13
8
8
10
10
3
3
8
8
1417
8
8
88
15
8
8
14
12
12
4
4
21
21
18
8
18
18
88
1717
17
17
24
24
13
13
23
23
22
22
3
27
14
12
3
3
8 8
8
8
1818
18
18
9
9
15
15
8
8
55
5
5
28
29
30
31
32
33
34
35
36
3636
39 39
117
4
4
5
55
34 3429
29
8
8
40
5
5
5
16
16
38 38
5
5
4
4
41
13
13
3131
3232
42
2828
4
30 30
7
7
6
6
6
44
11
5
58
6
21
21
44
11
11
1919
5
5
2020
2222
11
43
3535
4
4
44
3737
45
4
4
46
5 5 6
633
33
5
5
15
15
55
2727
5
4
14
14
2
2
12
12
3
3
3
3
5
5
5
99
1
1
4
4
44
2525
11
47
4
4
18
18
24
24
4
5
23 23
1010
2626
17
17
2
2
48
49
50
5152
53
54
55
56
Figure 4. Risk Sharing Networks in Darmang (top) and Pokrom (bottom) with walktrap com-
munity detection (colors, numbers).
RISK AVERSION IN NETWORKS 33
1
1
77
8
8
1
12
21 21
1
1
2
2
3
3
4
4
9
9
1
1
19
19
2020
13
13
4
4
25
1
1
1
3
3
7
77
11
11
26
7
7 4
4
27
28
1
1
1
17
14
14
3
4
4
4
18
5
29
1
1
4
4
17
17
5
5
1
4
4
3
3
16
16
1
1
15
22
22
18
18
16
24
24
30
3
3
1
1
6
6
1
12
3
3
2
2
4
4
23
23
6
15
15
10
10
1
1
3
3
2
2
4
4
7
7
77
7
7
2
2
3
3
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
17
1
2
2
5
2
2
2
18
19
20
21
5
9
9
22
7
5
5
22
5
5
11
5
5
22
2
2
5
2
6
6
2324
25
5
55
26
1
5
5
15
11
11
10
10
2
1
2
24 4
2
2
27
2
2
2
28
29
5
5
3
37
2
51
5
30
4
5
5
5
5
2
2
5
5
31
32
14
14
33
34
4
2
5
1
4
41
1
35
36
2
37
2
38
39
40
41
3
3
1313
42
43
55
5
5
2
2
2
5
5
88
2
2
5
4
3
44
15 15
45
16
16
1212
46
47
48
49
50
51
52
53
54
55
56
57 58
59
6061
62
63
64
65
66
67
68
69
70
71
72
73
Figure 5. Risk Sharing Networks in Oboadaka (top) and Konkonuru (bottom) with walktrap
community detection (colors, numbers).
34 OCTOBER 2019
Table 1—Dyadic Regression: Risk Sharing Network
(1) (2) (3) (4) (5)rsn rsn rsn rsn rsn
eta diff z -0.00991 -0.00761∗∗ -0.00239 -0.00197 0.00127(-1.11) (-3.21) (-0.32) (-0.70) (0.40)
deg sum 0.00878∗∗∗ 0.00705∗∗∗ 0.00705∗∗∗
(38.21) (25.67) (25.66)
fam 0.517∗∗∗ 0.419∗∗∗ 0.438∗∗∗
(32.22) (30.11) (29.08)
fam diff z -0.0197∗
(-2.39)
cons 0.267∗∗∗ -0.144∗∗∗ 0.171∗∗∗ -0.141∗∗∗ -0.144∗∗∗
(9.73) (-14.11) (7.55) (-10.87) (-10.97)N 71052 71052 71052 71052 71052
t statistics in parentheses
∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
B. Dyadic Regression
Table 1 reports the results from estimating equations 24, 25, 26, 27, and 28. I
include village level fixed effects in all dyad regression specifications, though this
does not effect the magnitudes estimated in any of the specifications. Reported t-
statistics have been adjusted by clustering at the dyad level. To make results more
interpretable, I transform risk aversions into z-scores before computing regressors.
This changes the interpretation just a bit. For example, β1 estimates the effect
of a one-standard deviation absolute difference in risk aversion.
Column 2 and 5 present my preferred specifications. This derives from an
empirical fact and my interpretation of the mechanism. First, less risk averse
individuals tend to be better connected than more risk averse individuals and
risk loving individuals. While we could control for the sum of risk aversion (and
RISK AVERSION IN NETWORKS 35
Table 2—Dyadic Regression: Community Network
(1) (2) (3) (4) (5)same comm same comm same comm same comm same comm
eta diff z -0.00668 -0.00161 -0.00343 0.000466 0.00145(-1.38) (-1.20) (-0.79) (0.32) (0.81)
comm size 0.00947∗∗∗ 0.00866∗∗∗ 0.00866∗∗∗
(12.91) (11.95) (11.95)
fam 0.224∗∗∗ 0.172∗∗∗ 0.178∗∗∗
(13.56) (12.17) (10.96)
fam diff z -0.00596(-0.68)
cons 0.135∗∗∗ -0.0878∗∗∗ 0.0933∗∗∗ -0.101∗∗∗ -0.102∗∗∗
(5.67) (-6.61) (4.26) (-7.92) (-7.90)N 71052 71052 71052 71052 71052
t statistics in parentheses
∗ p < 0.05, ∗∗ p < 0.01, ∗∗∗ p < 0.001
36 OCTOBER 2019
the sum of squared risk aversion), I control for this degree effect directly.
Across all specifications we see negative estimates of β1 or β1 + β4. However,
in columns 1 and 3, when the sum of degree is omitted from the model, the esti-
mates are small in magnitude and are not statistically significant (at any standard
confidence level). In contrast, controlling for degree in column 2 yields a negative
and significant (at the 1% level). I estimate a one standard deviation difference in
risk aversion leads a 0.76 percentage point reduction in the probability of linkage.
Family connections are also a strong determinant of linkage in the risk sharing
network. Across specifications 3, 4, 5, having a family connection is positively
associated with linkage in the risk sharing network (statistically significant at the
0.1% level). In column 5, family member dyads are 43.8 percentage points more
likely to form a risk sharing relationship as non-family members.
However, in columns 4 and 5, when I control for family connection and degree,
the estimate of β1 is once again insignificant (and positive in column 5). How-
ever, this may speak more to the mechanism of assortative matching. Similar to
Attanasio et al. (2012), we would expect assortative matching on risk aversion to
play a stronger role for more socially proximate individuals, who have more infor-
mation about each others preferences. In column 5, we have β1 + β4 = −0.0184,
statistically significant at the 5% level (χ2(1) = 6.61). Interpreting the coefficient,
between family members a one standard deviation difference in risk aversion re-
duces the probability of linkage by 18.4 percentage points.
Table 2 reports the results from re-estimating equations 24, 25, 26, 27, and 28
with the community network as the outcome.
The estimates of β1 are low in magnitude and vary in sign. None are statistically
significantly different than 0 at any standard significance levels. Likewise, in
column 5, β1 + β2 = −.005 is not significantly different than zero (χ2(1) = 0.34).
RISK AVERSION IN NETWORKS 37
C. Subgraph Generation Models with Types
Table 3—Links and Isolates Pooled Subgraph Generation Model: Risk Sharing Network with
Surveyed and Non-surveyed Network Members.
Feature Count Potential Sample size Coef. Std. Err.
Isolates:Not surveyed 36 96 631 0.375 0.0193
Surveyed 54 535 631 0.1009 0.012
Within links:Not surveyed 20 1133 49536 0.0177 0.0006
Surveyed 1367 35526 49536 0.0385 0.0009
Between links:Surveyed, not 221 12877 49536 0.0172 0.0006
Table 4—Links and Isolates Pooled Subgraph Generation Model: Community Network with
Surveyed and Non-surveyed Network Members.
Feature Count Potential Sample size Coef. Std. Err.
Isolates:Not surveyed 39 96 631 0.4062 0.0196
Surveyed 77 535 631 0.1439 0.014
Within links:Not surveyed 42 1133 49536 0.0371 0.0008
Surveyed 3108 35526 49536 0.0875 0.0013
Between links:Surveyed, not 703 12877 49536 0.0546 0.001
Table 3 presents the results of a links and isolates SUGM estimated with 2 types.
“Surveyed” reflects that individual was surveyed about their risk preferences, “not
surveyed” reflects the opposite. The risk sharing network used here is the risk
38 OCTOBER 2019
sharing network described above. The estimated features are links of all type
combinations (within and between types) and isolates of both types. The size
of the network is 631, but effective sample size varies by feature, as detailed
in Section XX (this remains true for Tables 4, 5, and 6). The estimates here
give us a baseline for how connected our surveyed individuals are to each other, a
natural baseline for comparing future estimates. Using this model, I estimate that
individuals who are surveyed about preferences tend to form links to each other at
a rate of 3.85%. Table 4 is a links and isolates model with 2 types estimated with
the community network as opposed to the risk sharing network. The network
arising from community detection tends to be denser than the preferred network:
I estimate that individuals who are surveyed about preferences tend to form links
to each other at a rate of 8.75%, more than twice the rate in the risk sharing
network.
Moving past the baseline model to the main model, Table 5 presents the results
of a links and isolates pooled SUGM estimated on the risk sharing network with
4 types. Less risk averse, more risk averse and risk loving reflect the risk pref-
erences of the individual, whereas not surveyed reflects that the individual was
not surveyed. We derive two main findings from this model. First, we see further
evidence of assortative matching on risk preference by less risk averse individuals.
Less risk averse agents form within-type links at a rate of 5.61% (compared to the
base rate of 3.85%). Second, we do not see the same kind of assortative matching
when looking at more risk individuals: I estimate more risk averse individuals
form within-type links at a rate of 2.99%, lower than both the base rate and
the rate at which less and more risk averse individuals form links between type
(3.60%).
Table 6 presents the results of a links and isolates pooled SUGM Mmdle esti-
RISK AVERSION IN NETWORKS 39
Table 5—Links and Isolates Pooled Subgraph Generation Model: Risk Sharing Network with
Risk Preferences
Feature Count Poten. S. size Coef. Std. Err.
Isolates:Less risk averse 22 236 631 0.0932 0.0116
More risk averse 19 217 631 0.0876 0.0113Risk loving 13 82 631 0.1585 0.0145
Not surveyed 36 96 631 0.375 0.0193
Within links:Less risk averse 421 7511 49536 0.0561 0.001
More risk averse 186 6223 49536 0.0299 0.0008Risk loving 33 814 49536 0.0405 0.0009
Not surveyed 20 1133 49536 0.0177 0.0006
Between links:Less, more risk averse 423 11738 49536 0.036 0.0008
Less risk averse, risk loving 163 4765 49536 0.0342 0.0008More risk averse, risk loving 141 4475 49536 0.0315 0.0008
Less risk averse, not surveyed 132 5879 49536 0.0225 0.0007More risk averse, not surveyed 66 5057 49536 0.0131 0.0005
Risk loving, not sureveyed 23 1941 49536 0.0118 0.0005
40 OCTOBER 2019
mated on the community network with 4 types. Using these estimates, I compare
the community network to the risk sharing network, controlling for the inflation
in degree. Controlling for the degree of connection, less risk averse agents are
less likely to encounter their own type in a risk sharing community as opposed to
their network neighborhood9
(43)R1,1
Rhp= 0.93 < 1.
Controlling for the degree of connection, more risk averse agents are more likely
to encounter their own type in a risk sharing community as opposed to in their
network neighborhood
(44)R2,2
Rhp= 1.05 > 1.
Controlling for the degree of connection, less and more risk averse agents are more
likely to encounter the other type within a risk sharing community as opposed to
their network neighborhood.
(45)R1,2
Rhp= 1.15 > 1.
D. Welfare Implications of Type Allocation
Scenarios
1) Worst case scenario: communities are entirely assortatively matched
9See here for the estimates of the intermediate ratios: Rhp =βCL,hp
βL,hp= 0.0875
0.0385= 2.27, R1,1 =
βCL,1
βL,1=
0.11890.0561
= 2.12, R2,2 =βCL,2
βL,2= 0.0713
0.0299= 2.38, and R1,2 =
βCL,1,2
βL,1,2= 0.0876
0.0360= 2.43.
RISK AVERSION IN NETWORKS 41
Table 6—Links and Isolates Pooled Subgraph Generation Model: Community with Risk Pref-
erences
Feature Count Poten. S. size Coef. Std. Err.
Isolates:Less risk averse 38 236 631 0.161 0.0146
More risk averse 22 217 631 0.1014 0.012Risk loving 17 82 631 0.2073 0.0161
Not surveyed 39 96 631 0.4062 0.0196
Within links:Less risk averse 893 7511 49536 0.1189 0.0015
More risk averse 444 6223 49536 0.0713 0.0012Risk Loving 59 814 49536 0.0725 0.0012
Not surveyed 42 1133 49536 0.0371 0.0008
Between links:Less, more risk averse 1028 11738 49536 0.0876 0.0013
Less risk averse, risk loving 373 4765 49536 0.0783 0.0012More risk averse, risk loving 311 4475 49536 0.0695 0.0011
Less risk averse, not surveyed 379 5879 49536 0.0645 0.0011More risk averse, not surveyed 248 5057 49536 0.049 0.001
Risk loving, not surveyed 76 1941 49536 0.0392 0.0009
42 OCTOBER 2019
2) No emergent properties: Same degree of assortative matching in network as
in community
3) Randomly sorted
4) Communities optimally sorted per the model above
For each of the above
• How does the allocation of types differ?
• How does this impact welfare?
RISK AVERSION IN NETWORKS 43
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RISK AVERSION IN NETWORKS 45
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46 OCTOBER 2019
Theoretical Appendix
A1. Risk Sharing in Communities
Feasibility of Risk Sharing. — Due to constraints 5, 6 and 7, budget constraints
bind at the community level. To see this, we sum up the two types using weights:
pc1i + (1− p)c2i ≤ θ
(1
N
N∑i=1
yi + yv
)+ pλ1
+ (1− θ)
(1
N
N∑i=1
yi + yv
)+ (1− p)λ2
≤ 1
N
N∑i=1
yi + yv + 0
N1c1i +N2c2i ≤N∑i=1
yi +Nyv.
Hence total consumption shocks to types 1 and 2 are bounded by total income
shocks and informal insurance is feasible.
Expected Utility. — Because shocks are normally distributed, expected utility
for both types is equivalent to maximizing the mean-variance representation This
can be seen in Sargent (1987).
E(U(c`i)) = E(c`i)−η`i2V ar(c`i)
Also note CARA is an increasing function in consumption, so in all states of
the world we consume all income and transfers available. So then we can write.
Expected consumption for type 1 is E(c1i) = λ1i and for type 2, E(c2i) = λ2i.
RISK AVERSION IN NETWORKS 47
Variance for the two types can be computed:
V ar(c1i) =
(θ
p
)2(σ2
N+ ν2
)
V ar(c2i) =
(1− θ1− p
)2(σ2
N+ ν2
).
So then we write expected utility
E(U(c1i)) = λ1i −η1i
2
(θ
p
)2(σ2
N+ ν2
)
and
E(U(c2i)) = λ2i −η2i
2
(1− θ1− p
)2(σ2
N+ ν2
).
For ease of notation, we define
σ2c =
σ2
N+ ν2.
and note that the utility of the the more risk averse agents when only idiosyncratic
risk is pooled is equal to
EU0 = −η2i
2
(1
N
N∑i=1
yi + yv
).
Solving the Lagrangian. — We construct the Lagrangian retaining constraints
4 and 7 (with a2 and a3 as multipliers, respectively) and incorporate the con-
48 OCTOBER 2019
sumption constraints into expected utlity.
L = λ1 −η1
2
θ2
p2σ2c+ a
(λ2 −
η2
2
(1− θ)2
(1− p)2σ2c +
η2
2σ2c
)+ b (pλ1 + (1− p)λ2)
The first order conditions are as follows:
∂L∂λ1
= 1 + bp = 0(A1)
∂L∂λ2
= a+ b(1− p) = 0(A2)
∂L∂θ
=−η1θσ
2c
p2+ a2
(η2(1− θ)σ2
c
(1− p)2
)(A3)
∂L∂a
= λ2 −η2
2
(1− θ1− p
)2
σ2c +
η2
2σ2c = 0(A4)
∂L∂b
=pλ1 + (1− p)λ2 = 0(A5)
Using FOC A1 we note that b = −1p . Likewise, using FOC A2 we note that
a = 1−pp . Rearranging FOC A4,
λ2 = −η2
2
(1−
(1− θ1− p
)2)
We rearrange FOC A5 and substitute in FOC A4:
λ1 = −1− pp
λ2 =1− pp
η2
2
(1−
(1− θ1− p
)2)
RISK AVERSION IN NETWORKS 49
Finally, we simplify FOC A3 to find θ:
η1θσ2c
p2=
1− pp
(η1(1− θ)σ2
c
(1− p)2
)(η1
η2
)(1− pp
)=
1− θθ
1
θ=
(η1
η2
)(1− pp
)+ 1
θ =pη2
(1− p)η1 + pη2
Thus only if either η1 = 0 (type 1 is risk neutral, which we’ve assumed is not
true) or p = 1, θ = 1. Hence, covariate risk will not taken on fully by the less risk
averse agents. Note that
(1− θ)2 =
(1− pη2
(1− p)η1 + pη2
)2
=
(1− (1− p)η1
(1− p)η1 + pη2
)2
=(1− p)2η2
1
((1− p)η1 + pη2)2
So then
λ2 = −η2
2
(1− η2
1
((1− p)η1 + pη2)2
)
Value Functions. — We compute the value functions for type 1 and type 2
individuals.
V1(p, η1, η2) = E(U(c1i)|θ∗(p), λ∗1(p))
50 OCTOBER 2019
= λ∗1 −η1
2
(θ∗(p)
p
)2
σ2c
= λ∗1 −η1
2
(pη2
((1− p)η1 + pη2)p
)σ2c
= λ∗1 −η1
2
(η2
((1− p)η1 + pη2)
)σ2c
=η1
2
(1− pp
)(1−
(η1
(1− p)η1 + pη2
)2)− η1
2
(η2
(1− p)η1 + pη2
)2
σ2c
V2(p, η1, η2) = E(U(c2i)|θ∗(p), λ∗2(p))
= λ∗2(p)− η2
2
(1− θ∗(p)
1− p
)2
= λ∗2(p)− η2
2
(1− pη2
(1−p)η1+pη2
1− p
)2
σ2c
= λ∗2(p)− η2
2
((1− p)η1 + pη2 − pη2
(1− p)((1− p)η1 + pη2)
)2
σ2c
= λ∗2(p)− η2
2
((1− p)η1 + pη2 − pη2
(1− p)((1− p)η1 + pη2)
)2
σ2c
= λ∗2(p)− η2
2
((1− p)η1
(1− p)((1− p)η1 + pη2)
)2
σ2c
= λ∗2(p)− η2
2
(η1
(1− p)η1 + pη2
)2
σ2c
= − η2
2
(1− η2
1
((1− p)η1 + pη2)2
)− η2
2
(η1
(1− p)η1 + pη2
)2
σ2c
=η2
2
(1 +
(η1
(1− p)η1 + pη2
)2 (1 + σ2
c
))
RISK AVERSION IN NETWORKS 51
More. —
pη2 + (1− p)η1 < pη2 + (1− p)η2
= η2
so then
1
pη2 + (1− p)η1>
1
η2
pη2
pη2 + (1− p)η1>pη2
η2= p
A2. Planner’s Problem
Some useful equations:
(A6)
∂V1(q)
∂q=
((1− q
1
)η2
1 + σ2cη
22
)((η2 − η1)η2
((1− q)η1 + qη2)3
)− η2
2
2q2
(1−
(η1
(1− q)η1 + qη2
)2)
(A7)∂V2(q)
∂q= −
(η2η1(η2 − η1)
((1− q)η1 + qη2)3
)(σ2c − 1
)
(A8)∂V2(1− q)
∂q=
(η2η1(η2 − η1)
(qη1 + (1− q)η2)3
)(σ2c − 1
)
52 OCTOBER 2019
∂V1(q)
∂q=
(A9)
η2
2
(− 1
q2
(1−
(η1
(1− q)η1 + qη2
)2)
+
(1− qq
)(2η2
1(η2 − η1)
((1− q)η1 + qη2)3
))
− η2
2
(−2η2
2(η2 − η1)
((1− q)η1 + qη2)3
)σ2c
=
(1− q
1
)η2
1
((η2 − η1)η2
((1− q)η1 + qη2)3
)+ σ2
cη22
((η2 − η1)η2
((1− q)η1 + qη2)3
)(A10)
− η22
2q2
(1−
(η1
(1− q)η1 + qη2
)2)
=
((1− q
1
)η2
1 + σ2cη
22
)((η2 − η1)η2
((1− q)η1 + qη2)3
)− η2
2
2q2
(1−
(η1
(1− q)η1 + qη2
)2)(A11)
Empirical Appendix
B1. Data
Risk Preferences and Prospect Theory. — To work in the prospect theory
framework, it becomes important to work with a more general form of the expo-
nential utility function. This more general function will be constructed to allow
for risk aversion and risk loving behavior and to take into account individuals
RISK AVERSION IN NETWORKS 53
Loss Loving
Loss Neutral
Loss Averse
−1000
−500
0
−200 −100 0 100 200
Income
Util
ityRisk Loving in Gains, Risk Averse in Losses
Unicorns
Loss Loving
Loss Neutral
Loss Averse
−1000
−500
0
−200 −100 0 100 200
Income
Util
ity
Risk Averse Agents
Figure B1. Utility functions of individuals with various types of risk preferences (scaled to
be equal in the gains domain).
who have different preferences in the gains and losses domain.
(B1) u(c) =
1−e−ηc|κ|η if c > 0
c if c = 0
1−e−κηcκη if c < 0
Leaving aside risk neutrality, using a set of four hypothetical gambles we can
describe 12 different types of agents. Agents are (risk averse in gains, risk loving
in gains) × (risk averse in losses, risk loving in losses) × (loss averse, loss neutral,
loss loving). In general, we lay aside loss aversion and consider four main types
of individuals. First, risk averse individuals who are risk averse in both gains and
losses (η > 0, κ > 0). Second, risk loving individuals who are risk loving in both
gains and losses (η < 0, κ > 0). Third, prospect theory style individuals who
exhibit an S-shaped curve similar to the one discussed in Kahneman and Tversky
(1979) (η > 0, κ < 0). Fourth, “unicorns” who are risk loving in gains but risk
54 OCTOBER 2019
Loss Loving
Loss Neutral
Loss Averse
−200
0
200
−200 −100 0 100 200
Income
Util
ity
Risk Loving Agents
Loss Loving
Loss Neutral
Loss Averse
−200
−100
0
100
200
−200 −100 0 100 200
Income
Util
ity
Risk Averse in Gains, Risk Loving in Losses
Prospect Theory Agents
Figure B2. Utility functions of individuals with various types of risk preferences (scaled to
be equal in the gains domain).
averse in losses (η < 0, κ < 0).10 These potential preferences are plotted in figure
B2.
The first two menus presented are in the gains domain. In the first menu, the
risky gamble YB is held fixed while increasing the sure payment. In the second,
the sure payment is held fixed while reducing the upside of the gamble. We
assume that if an individual reaches a point of indifference between two gambles,
we assign them to the midpoint between the two gambles. Hence, if the mean
differs, we take the average of the mean of the two gambles and assign this value to
the point of indifference. If the variance differs, we take the average of variances
and assign this value to the point of indifference. The second two menus are
reflections of the first two onto the domain of losses.
We compute either η or κη (the combined coefficient of risk aversion in the
domain of losses) for each lottery and individual. We have two sets of two indif-
10These unicorns may not be unicorns in situations with small probabilities, as in Kahneman andTversky’s fourfold pattern. This is not our context.
RISK AVERSION IN NETWORKS 55
Prospect TheoryRisk Loving
Risk AverseUnicorns
−0.0050
−0.0025
0.0000
0.0025
0.0050
−0.0050 −0.0025 0.0000 0.0025 0.0050Risk Aversion in Gains
Ris
k A
vers
ion
in L
osse
s
n
50
100
150
Risk Aversion by Domain
Figure B3. Distribution of individual preference parameters from CARA utility. x-axis cor-
responds to η, y-axis corresponds to κη.
ference points, sp to combine these into measures of risk aversion, we average η
and κη over individuals, and use these averages ηi κηi and individual parameters.
We then are able to find the value of κi, taking κηi/ηi.
We can still represent expected utility as a mean variance utility function in
the domain of gains and the domain of losses. In the domain of losses, we write
(B2) E(Y )− κη
2V (Y )
56 OCTOBER 2019
I look at a menu of choices between hypothetical gambles. YA is a set of sure
payments and YB is a set of risky income sources. For mathematical tractability,
we assume YB is normally distributed. In the domain of gains, respondents are
able to choose between these will be indifferent between the two when
(B3) E(YB)− η
2V (YB) = YA
Observing this indifference point, we can write
(B4) η =2(E(YBg)− YAg)
V (YBg)
and recover the coefficient of absolute risk aversion. Likewise, in the domain of
losses, at the indifference point
(B5) E(−YBl)−κη
2V (−YBl) = −YAl.
We write
(B6) κη =2(YAl − E(YBl))
V (YBl)
So then κ is the ratio of risk aversion in the domain of gains to risk aversion in
the domain of losses. This simplifies to the ratio of the respective risk premiums.
(B7) κ =E(YBg)− YAgYAl − E(YBl)
=RPgRPl
Within individuals, measures of risk aversion are highly correlated. For exam-
RISK AVERSION IN NETWORKS 57
ple, in the gains domain the η’s between the two lotteries correlates at a rate of
0.703. Likewise, κη’s correlate at a rate of 0.684. Between the domain of gains
and losses, correlations are lower, but still on the order of about 0.4. See Table
B1. Of those with full data 313 are risk averse, 127 are prospect theory, 46 are
risk loving, and 18 are unicorns. The distribution of preferences is plotted in
Figure B3.
Pref. Parameterηa ηb κηc κηd
ηa 1.000 0.703 0.400 0.436ηb 1.000 0.405 0.409κηc 1.000 0.684κηd 1.000
Table B1—Pearson correlations between measures of risk aversion based on menus of gambles
in the domain of gains and domain of losses.
B2. Community Detection
Other Approaches. — While the walktrap algorithm features good proper-
ties for community detection, it is by no means the only option. One appealing
approach to assign households to communities is edge betweenness community
detection. This algorithm is appealing because it takes advantage of information
bottlenecks in networks, connecting naturally to the types of asymmetric informa-
tion problems that constrain risk sharing (Girvan and Newman, 2003). For more
on edge betweenness, see B.B2. For large networks (like mobile money transac-
tion networks) there may be better methods from the perspective of computation
speed. However, detected communities from a given algorithm may not scale well
to larger networks if community size grows with network size. An iterated min
cut algorithm may fall closer to the conditions that create ex post risk sharing
58 OCTOBER 2019
islands Ambrus et al. (2014).
Edge Betweenness. — Another intuitive approach to finding communities hinges
on information in networks. Edge betweenness measures how central an edge is
in a network by counting which nodes it lies between. In particular, if edge kl
lies on the shortest path from i to j edge kl is awarded a unit of edge betweeness.
Summing up all of these awards from all pairs of nodes i and j, we get the
edge betweeness for edge kl. We can think of edges with high betweenness as
relationships in the risk sharing network where both individuals knowledge of
the other individual and the individuals beyond them comes directly through
that individual. There is a great deal of potential to broker information over
these links but because of frictions due to adverse selection, hidden income, and
various types of moral hazard, individuals will be hesitant to share risk with those
who they do not have good information about.
The algorithm is named because it calculates the betweenness centrality of all
edges in the network and deletes those edges with highest centrality. Edge be-
tweenness centrality is a measure of how central an edge is in a network based
on who relies on that edge connect to other portions of the network. In order to
compute this measure, compute all shortest paths between nodes on a network.
Then, count of the number of shortest paths passing through each edge of interest
(in the case multiple paths tie for shortest path for two nodes, a partial count is
awarded across the edges in these paths). Intuitively, betweenness centrality is
often used as a measure the potential for brokerage. In this case, we can think
of information flow being costly in the link that has high betweenness centrality
because of the gatekeeper on each side. This would create a specific rationale for
these communities as risk sharing units, then. One issue with edge betweenness is
that weighted versions of the algorithm may not interpret weights in an intuitive
RISK AVERSION IN NETWORKS 59
manner for our application. In particular larger capacity connections are lower
cost and hence have more least cost paths contributing to betweenness. This will
cause these edges to be cut early. Initially I will solve this by supplying inverse
weights to the algorithm. At the start of the algorithm, all nodes in a partic-
ular component are assigned to the same community. Every time a component
breaks into two with the deletion of an edge, the communities membership is
reassigned for the nodes in the broken component. After initially computing all
edge betweenness centralities, the algorithm works in two steps:
1) Delete the edge with the highest betweenness centrality
2) Recompute the betweenness centrality of remaining edges
This two step process continues until all edges are deleted and thus all nodes re-
side in their own component and hence community. To choose a final community
assignment, the algorithm cuts the dendrogram using a tuning statistic. For this,
I have two possible solutions. First, the off the shelf choice in network science
is modularity, which measures the internal quality of communities assigned (see
Appendix B.B2 for details). The algorithm cuts of the dendrogram at the as-
signment with the maximum value of the tuning statistic (Girvan and Newman,
2003). In previous unpublished work, I used edge betweenness community detec-
tion to detect risk sharing islands using data from southern India. Alternatively,
a regression coefficient measuring the degree of risk sharing in the community as-
signment could be used to tune the assignment. I leave discussion of this second
option until tests of risk sharing have been discussed.
Modularity. — To compute modularity let ki and kj be the degrees of nodes i
and j, respectively. Let m be the number of edges in the graph. The expected
number of edges between i and j from this rewiring is equal to kikj/(2m− 1) ≈
60 OCTOBER 2019
kikj/2m (2m since each link has two “stubs”, so to speak). I can then compare
this expected number of links between i and j to the actual connections: letting
Aij be the ijth entry of the matrix, I take the difference these two numbers:
Aij −kikj2m
I can interpret this as connections over expected conditional on node pair de-
grees. Then, letting ci be the community membership of node i, connections over
expectation are weighted by the function δ:
δ(ci, cj) =
1 if ci = cj
0 otherwise
Finally I aggregate to the graph level and normalize by twice the number of links
present:
Q =1
2m
∑ij
[Aij −
kikj2m
]δ(ci, cj)
This serves as an easily computable and straightforward measure of the internal
quality of communities (Newman, 2011).
RISK AVERSION IN NETWORKS 61
Additional Tables
Table C1—Pooled Subgraph Generation Model with Links and Isolates and Type: Surveyed.
Network size = 631.
Feature Count Potential Sample size Coef. Std. Err.
Isolates:Surveyed 50 535 631 0.0935 0.0116
Not surveyed 35 96 631 0.3646 0.0192
Within links:Surveyed 1535 35526 49536 0.0432 0.0009
Not surveyed 20 1133 49536 0.0177 0.0006
Between links:Surveyed, not 247 12877 49536 0.0192 0.0006
Table C2—Pooled Subgraph Generation Model using Community Graph with Links and Iso-
lates and Type: Surveyed. Network size = 631.
Feature Count Potential Sample size Coef. Std. Err.
Isolates:Surveyed 71 535 631 0.1327 0.0135
Not surveyed 39 96 631 0.4062 0.0196
Within links:Surveyed 3411 35526 49536 0.096 0.0013
Not surveyed 38 1133 49536 0.0335 0.0008
Between links:Surveyed, not 653 12877 49536 0.0507 0.001
62 OCTOBER 2019
Table C3—Pooled Subgraph Generation Model with Links and Isolates and Type: Prefer-
ences. Network size = 631.
Feature Count Potential Sample size Coef. Std. Err.
Isolates:Less risk averse 21 236 631 0.089 0.0113
More risk averse 19 217 631 0.0876 0.0113Risk loving 10 82 631 0.122 0.013
Not surveyed 35 96 631 0.3646 0.0192
Within links:Less risk averse 471 7511 49536 0.0627 0.0011
More risk averse 203 6223 49536 0.0326 0.0008Risk Loving 36 814 49536 0.0442 0.0009
Not surveyed 20 1133 49536 0.0177 0.0006
Between links:Less, more risk averse 478 11738 49536 0.0407 0.0009
Less risk averse, risk loving 192 4765 49536 0.0403 0.0009More risk averse, risk loving 155 4475 49536 0.0346 0.0008
Less risk averse, not surveyed 146 5879 49536 0.0248 0.0007More risk averse, not surveyed 75 5057 49536 0.0148 0.0005
Risk loving, not surveyed 26 1941 49536 0.0134 0.0005
RISK AVERSION IN NETWORKS 63
Table C4—Pooled Subgraph Generation Model using Community Graph with Links and Iso-
lates and Type: Preferences. Network size = 631.
Feature Count Potential Sample size Coef. Std. Err.
Isolates:Less risk averse 36 236 631 0.1525 0.0143
More risk averse 21 217 631 0.0968 0.0118Risk loving 14 82 631 0.1707 0.015
Not surveyed 39 96 631 0.4062 0.0196
Within links:Less risk averse 991 7511 49536 0.1319 0.0015
More risk averse 475 6223 49536 0.0763 0.0012Risk Loving 64 814 49536 0.0786 0.0012
Not surveyed 38 1133 49536 0.0335 0.0008
Between links:Less, more risk averse 1171 11738 49536 0.0998 0.0013
Less risk averse, risk loving 379 4765 49536 0.0795 0.0012More risk averse, risk loving 331 4475 49536 0.074 0.0012
Less risk averse, not surveyed 356 5879 49536 0.0606 0.0011More risk averse, not surveyed 229 5057 49536 0.0453 0.0009
Risk loving, not surveyed 68 1941 49536 0.035 0.0008
64 OCTOBER 2019
Additional Figures
0.00
0.25
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0.75
1.00
0.00 0.25 0.50 0.75 1.00Proportion of Covariate Risk Taken on by
Less Risk Averse Agents in Smaller Community
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of C
ovar
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Ris
k Ta
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on b
y Le
ss R
isk
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rse
Age
nts
in L
arge
r C
omm
unity
Size of smaller community
3035404550
Figure D1. A Risk Management Frontier: Proportion of Covariate Risk Taken on by Less
Risk Averse Agents in Communities. From top left to bottom right, type 1 agents move from
larger community to smaller.
RISK AVERSION IN NETWORKS 65
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Figure D2. Risk Sharing Networks in Darmang (left) and Pokrom (right) with spouses (color
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(color = spouse type) and household numbers.
66 OCTOBER 2019
1 Steps, Modularity= 0.14
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Figure D4. Detected communities using the Walktrap algorithm for a gift network in the
village of Darmang. Panels depict walks of different lengths, starting at length 1 and
ending at length 8. Within this range, as walks lengthen, the number of communities detected
falls from 13 to 7.